[LINK] One for the Numerologists and Mystics...

[sylvano]

Happy New Year, everyone!

One of the perplexity results doesn't add up:

**Sum of Consecutive Primes**
Interestingly, 2025 can be represented as the sum of five consecutive
prime numbers: 397 + 401 + 409 + 419 + 421 = 2025[7]. This connection
to prime numbers adds to its number-theoretical significance.


397 + 401 + 409 + 419 + 421 = 2047


Sylvano

On Tue, 2024-12-24 at 21:25 +1100, Antony Barry wrote:
> On Tue, Dec 24, 2024 at 9:05 PM David <dlochrin at aussiebb.com.au>
> wrote:
> 
> >      *  2025, itself is a square, i.e. 45 x 45 = 2025
> >      *  it is also a product of two-squares, i.e. 9² x 5² = 2025
> >      *  it is also the sum of 3-squares, i.e. 40²+ 20²+5²= 2025
> >      *  and remarkably it is also the sum of the cubes, of all the
> > single
> > digits, from 1 to 9, i.e.1³+2³+3³+4³+5³+ 6³+7³+8³+9³= 2025.
> 
> I asked Perplexity AI is there was anything interesting about 2025
> with the
> following result -
> 
> 2025 indeed possesses several interesting properties that make it
> noteworthy in number theory:
> 
> ## Mathematical Properties
> 
> **Perfect Square**
> 2025 is a perfect square, specifically 45^2[1][4]. This property
> alone
> makes it significant in number theory, as square numbers often
> exhibit
> unique characteristics.
> 
> **Sum of Consecutive Odd Numbers**
> 2025 can be expressed as the sum of five consecutive single-digit odd
> natural numbers: 1 + 3 + 5 + 7 + 9[3]. This property is particularly
> interesting as it connects 2025 to other mathematical sequences.[Note
> by
> Tony: Perplexity seems to have got confused!]
> 
> **Unique Squaring Property**
> 2025 is probably the only positive square which remains a square when
> adding 1 to each digit (resulting in 3136) as well as when increasing
> just
> the first digit (resulting in 3025)[2]. This unusual characteristic
> sets
> 2025 apart from other numbers.
> 
> ## Number Theoretical Aspects
> 
> **Prime Factorization**
> The prime factorization of 2025 is 3^4 * 5^2[1][6]. This
> decomposition into
> prime factors is fundamental in number theory and contributes to many
> of
> its other properties.
> 
> **Divisibility**
> 2025 has 15 divisors[1], making it a highly composite number. It is
> also a
> tau number, as it is divisible by the number of its divisors[2].
> 
> **Special Number Classifications**
> 2025 falls into several interesting number categories:
> - It is a Harshad number (divisible by the sum of its digits)[2].
> - It is a Curzon number[2].
> - It is an apocalyptic number[2].
> - It is a gapful number (divisible by the number formed by its first
> and
> last digit)[2].
> 
> ## Unique Representations
> 
> **Sum of Squares**
> 2025 can be written as a sum of two non-zero squares in only one way:
> 27^2
> + 36^2[2]. This property relates to the Pythagorean theorem and
> Fermat's
> theorem on sums of two squares.
> 
> **Sum of Consecutive Primes**
> Interestingly, 2025 can be represented as the sum of five consecutive
> prime
> numbers: 397 + 401 + 409 + 419 + 421 = 2025[7]. This connection to
> prime
> numbers adds to its number-theoretical significance.
> 
> ## Conclusion
> 
> These properties collectively make 2025 a number of considerable
> interest
> in number theory. Its status as a perfect square, its unique
> representations as sums of consecutive numbers (both odd and prime),
> and
> its various special number classifications demonstrate that 2025
> possesses
> a rich set of mathematical characteristics worthy of study in number
> theory.
> 
> Sources
> [1] Properties of the number 2025 - Number Empire
> https://www.numberempire.com/2025
> [2] Properties of number 2025 - Numbers Aplenty
> https://www.numbersaplenty.com/2025
> [3] 25 (number) - Wikipedia https://en.wikipedia.org/wiki/25_(number)
> [4] Number 2025 - Math Tools https://math.tools/numbers/2025
> [5] Properties of natural numbers such that there is a "very large
> largest
> ...
> https://mathoverflow.net/questions/101463/properties-of-natural-numbers-such-that-there-is-a-very-large-largest-number-w
> [6] Prime factors of 2025 - Math Tools
> https://math.tools/numbers/prime-factors/2025
> [7] Do you know this The number 2025 can be represented ... - Threads
> https://www.threads.net/@data.science.beginners/post/DD7WsWTTuaP
> [8] what is special about the number 2025? - hellosunday.ca
> https://hellosunday.ca/Vierte4/what-is-special-about-the-number-2025
> 
> Tony
> 
> > 
-- 

sylvano
sylvano at gnomon.com.au